The lifespans of meerkats in a particular zoo are normally distributed. The average meerkat lives $13.4$ years; the standard deviation is $3.1$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a meerkat living between $7.2$ and $22.7$ years.
$13.4$ $10.3$ $16.5$ $7.2$ $19.6$ $4.1$ $22.7$ $99.7\%$ $95\%$ $2.35\%$ $2.35\%$ We know the lifespans are normally distributed with an average lifespan of $13.4$ years. We know the standard deviation is $3.1$ years, so one standard deviation below the mean is $10.3$ years and one standard deviation above the mean is $16.5$ years. Two standard deviations below the mean is $7.2$ years and two standard deviations above the mean is $19.6$ years. Three standard deviations below the mean is $4.1$ years and three standard deviations above the mean is $22.7$ years. We are interested in the probability of a meerkat living between $7.2$ and $22.7$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the meerkats will have lifespans within 3 standard deviations of the average lifespan. It also tells us that $95\%$ of the meerkats will have lifespans within 2 standard deviations of the mean. That leaves $99.7\% - 95\% = 4.7\%$ of meerkats between 2 and 3 standard deviations of the mean, or $2.35\%$ on either side of the distribution. The probability of a particular meerkat living between $7.2$ and $22.7$ years is ${95\%} + \color{orange}{2.35\%}$, or $97.35\%$.